Participants in Geurts and van der Silk’s (2005) experiments on reasoning with gen- eralized quantifiers rightly judged valid monotonicity inferences to be correct with a mean success varying between 36% and 96%. We will now use our collected remarks on monotonicity above to create weights that do this variation justice.
We noticed that the directionality of the quantifiers involved has an impact on the cognitive difficulty of iterated quantifiers in the sense that downwardness increases cost. Further, cost is increased if the two quantifiers do not have the same directionality (harmony) and if the first quantifier switches the directionality of the second (in the QI- fragment, this is the case whenever the first quantifier is right-side downward entailing). Cost decreases however if one of the informative quantifier expressions is in the major premise (ALLandNOT). All factors are normalized s.t. assigned values are between 0
and 1 (directionality and hierarchy allow for values 0, 0.5, and 1 while hierarchy and switch only allow for values 0 and 1). The results of this are summed up in table 8. Where there are two values, the first one holds when the first quantifier isNOorALL
and the second one if not (note that the second quantifier in the QI-fragment is fixed to
MORE THANorFEWER THAN).
We realized further above that monotonicity inferences on iterated quantifiers need not be harder than those on single quantifiers. We will define the base cost of a mono-
Table 8:Costs assigned to combinatorial monotonicity profiles according to upwardness, harmony and switch. ”Negative points are gathered that state to which factor an inference relates to the basic cost of 15.
CMP Up/Down Harmony Switch Hierarchy Overall
↑↑ 0 0 0 0/1 0/1
↑↓ 0.5 1 0 0/1 1.5/2.5
↓↑ 0.5 1 1 0/1 2.5/3.5
↓↓ 1 0 1 0/1 2/3
tonicity inference on iterated quantifiers to be 15, reflecting that they need not be harder but that their difficulty increases faster when the aggravating factors discussed above come into play. The cost of an inference is then given by multiplying the basic cost 15 with the factor in table 8. The weights according to this procedure can be seen in table 9. For example, Mon↓↑gets 0.5 directionality points, 1 harmony point, 1 switch point and
Table 9:Weights for the inference rules in the QI-fragment as computed above. Numbers are rounded up.
Inference Mon↑↑ Mon↑↓ Mon↓↑ Mon↓↓
Complexity 0/15 23/38 38/53 30/45
1 hierarchy point, if the first quantifier isNO. That sums up to 2.5 and 3.5, respectively, yielding complexity 15·2.5=38 and 15·3.5=53, respectively for Mon↓↑-inferences (rounded to next integer). Hypothesizing that this model can be readily extended to left-side inferences, we see that their weights are:
Inference ↑↑Mon ↑↓Mon ↓↑Mon ↓↓Mon Complexity 0/15 23/38 23/38 15/30
Recall that they are meant to be generally easier because left-side monotonicity entail- ments cannot switch directions. This extension of the model will later lead to interesting predictions.
6
Natural Logic at Work
We think that this logic is adequate for the task of modeling the syllogistic- and the QI-fragment. Going back to our natural logic roadmap (chapter 3.2) – the inference rules presented account for all of the valid inferences in the QI-fragment, all syllogisms that are valid in Aristotelian logic or predicate calculus or both and even defines some others as good (e.g. AO3O, see table 4). Our cognitive goals imposed two constraints on the logic: firstly, apart from the logic having to account for the whole fragment, inference
rules should be informed and justified by semantical reasoning or empirical results from psychology. We have extensively done this in chapter 5. Secondly, we imposed a simplicity constraint to make away with rules that are not necessary to account for the whole fragment. We have seen this in our decision not to include inferences based on smoothness above. It follows immediately, that our logic is incomplete, but completeness has never been the goal. It is even doubtful that it makes sense to apply the traditional metalogical vocabulary to NQL: both notions of soundness and completeness are relative to a fixed semantics. And while we have given semantics for generalized quantifiers above, we also noted that when it comes to their inferential properties, we prefer to remain deliberately unclear about their semantics – we prefer interpretation-independent inferences over interpretation-dependent ones as far as this is possible. We had argued further above that we prefer talk aboutgoodand badinferences over talk aboutvalid andinvalidones as in chapter 2, we have seen a variety of normative standards none of which has a priori priority over the others. One possible consequence of our cognitive motivation is thus to disregard the traditional metalogical vocabulary for the analysis of NQL.15We will now get to the last step of our roadmap and evaluate our model against our initial goals that seem more fitting than an analysis of its metalogical properties. Before we will get to this evaluation, however, we will have to make some remarks on how to interpret our results and the relationship between the complexity of a rule and mean success of reasoners in a task.